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Iso, Yuusuke

Convergence of a semi-discrete finite difference scheme applied to the abstract Cauchy problem on a scale of Banach spaces. (English) Zbl 1236.65060

Proc. Japan Acad., Ser. A 87, No. 7, 109-113 (2011).
Summary: We show convergence of a finite difference scheme to solve the abstract Cauchy problem on a scale of Banach spaces which includes that for Kowalevskaya’s system. We show convergence of consistent difference schemes even for unstable cases.

Cited in 1 Document

MSC:

65J08 Numerical solutions to abstract evolution equations
65J10 Numerical solutions to equations with linear operators
47D06 One-parameter semigroups and linear evolution equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems

Keywords:

finite difference scheme; Cauchy-Kowalevskaya’s theorem; ill-posed problems; multiple-precision arithmetic; numerical example; semidiscretization; convergence; abstract Cauchy problem; Banach spaces
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\textit{Y. Iso}, Proc. Japan Acad., Ser. A 87, No. 7, 109--113 (2011; Zbl 1236.65060)
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References:

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[2] P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. 9 (1956), 267-293. · Zbl 0072.08903
[3] T. Nishida, A note on a theorem of Nirenberg, J. Differential Geom. 12 (1977), 629-633. · Zbl 0368.35007
[4] L. V. Ovsjannikov, Singular operator in the scale of Banach spaces, Dokl. Akad. Nauk SSSR 163-4 (1965), 819-822. (Eng. Trans. Soviet Math. Dokl. 6 (1965), 1025-1028). · Zbl 0144.39003
[5] T. Yamanaka, Note on Kowalevskaja’s system of partial differential equations, Comment. Math. Univ. St. Paul 9 (1961), 7-10. · Zbl 0122.35302
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